We know (Pesachim 56b) that Baruch Sheim, at least according to one explanation, is a tefillah Moshe heard from the Malachim. Thus, we only say it out loud when we're on the level of Malachim.

Yet, we (by that I mean the minhag in my community* - not sure whether this is true everywhere) say Baruch Sheim out loud after messing up a bracha beyond the point of "lamdeini chukecha." If we're not on the level of malachim, why should we be allowed to do so?

*Note that my community is primarily formed of talmidim of R' Weinberg of Ner Yisrael fame.

IIRC RJBS would say BSKMLV to himself on Yom Kippur if saying Shema alone. Seemingly he felt the whole outloud/quiet business is just part of Tefillah beTzibbur, so there'd be no preference in your case.
– Double AA♦Jun 24 '16 at 15:47